By Henk Broer, Igor Hoveijn, Gerton Lunter, Gert Vegter
The authors give some thought to purposes of singularity thought and desktop algebra to bifurcations of Hamiltonian dynamical structures. They limit themselves to the case have been the subsequent simplification is feasible. close to the equilibrium or (quasi-) periodic answer into account the linear half permits approximation by means of a normalized Hamiltonian approach with a torus symmetry. it's assumed that relief by way of this symmetry results in a procedure with one measure of freedom. the quantity makes a speciality of such aid tools, the planar relief (or polar coordinates) process and the relief through the power momentum mapping. The one-degree-of-freedom procedure then is tackled by means of singularity thought, the place computing device algebra, specifically, Gröbner foundation thoughts, are utilized. The readership addressed includes complex graduate scholars and researchers in dynamical systems.
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Nondegeneracy conditions) This is possible provided that the coefficient of the third-order terms (in x, y) are nonzero. 9). t. from H r . This morphism should respect the Z2 symmetry (x, y) → (x, −y). 11 in Sect. 2. Armed with the knowledge that φ exists we set out to compute it, using the following iterative approach. 11) H r ◦ φk = x(x2 + y 2 ) + O(|x, y|k+3 ) for some k. To find φk with k = k + 1 we set φk = φk + αi ti , where {ti } span the space of Z2 -equivariant terms in x, y of degree k .
This gives the natural set-up for the aforementioned perturbation problem. 2 See [GG73]. Necessity of this condition is immediate by considering deformations of the form H u (x, y, 0) + c1 g(x, y) for arbitrary (symmetric) g; see [Mar82, Prop. 2]. 3. Spring-pendulum in 1:2-resonance ✻ m1 x1 x2 m2 Fig. 1 The spring-pendulum with its axis of symmetry See [BCKV93] for a general discussion. 3. 3 Spring-pendulum in 1:2-resonance This chapter ends with the application of the planar reduction method to the spring-pendulum system.
2). g. Fig. 2, center picture. In Fig. 5 the pole is blown up to a circle, so that there the connection seems to be heteroclinic. 3. Spring-pendulum in 1:2-resonance out ∪-shaped and ∩-shaped paths, respectively. As the system moves away from resonance, one of these paths gets wider while the other gets narrower, until at bifurcation, the narrow one coincides with the short periodic orbit. After the bifurcation only one long periodic orbit remains (which is stable); the short periodic orbit has also become stable.
